Customer-service agents at American Airlines narrowly rejected a union’s bid to represent them in collective bargaining with the company.

The airport and reservations-center agents voted against representation by the Communications Workers of America by 3,052 to 2,902, or 51 percent to 49 percent. About 76 percent of eligible workers voted, according to results released Tuesday.

The union said it lost votes because American cut about 2,000 agent jobs after filing for bankruptcy protection in November 2011. Spokeswoman Candice Johnson said departing workers had to give up job-recall rights to get severance payments, which made them ineligible to vote. “Those were union votes,” she said.

American acknowledged that about 900 agents who took early-out bonuses gave up their recall rights, and those who left before the election couldn’t vote. The airline countered, however, that it wanted about 800 people hired after the union filed its election petition to vote, but it lost a ruling on the issue from the National Mediation Board, which oversees union-representation elections.

The union could launch another organizing campaign, but rules approved by Congress last year would make it harder to force an election. The issue also could come up again if American parent AMR Corp. merges with US Airways, whose agents are represented by CWA and the Teamsters.

American went all the way to the U.S. Supreme Court in a failed bid to block the election.

American argued that the union should have been required to show support from 50 percent of eligible workers before an election under a change in voting rules approved by Congress in February 2012. But the union argued that it only needed 35 percent support to force a vote under rules that were in effect when it filed for an election, and the courts upheld that view.

The agents are the largest bloc of nonunion workers at American. The airline’s pilots, flight attendants and ground workers are represented by other unions.